American Institute of Mathematical Sciences

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  2010, 17: 104-121. doi: 10.3934/era.2010.17.104

Notes on monotone Lagrangian twist tori

Yuri Chekanov 1, and  Felix Schlenk 2,

1. 

Moscow Center for Continuous Mathematical Education, B. Vlasievsky per. 11, Moscow 121002, Russian Federation
2. 

Institut de Mathématiques, Université de Neuchâtel, Rue Émile Argand 11, CP 158, 2009 Neuchâtel, Switzerland

Received  April 2010 Revised  July 2010 Published  October 2010

We construct monotone Lagrangian tori in the standard symplectic vector space, in the complex projective space and in products of spheres. We explain how to classify these Lagrangian tori up to symplectomorphism and Hamiltonian isotopy, and how to show that they are not displaceable by Hamiltonian isotopies.
Keywords: tori., Lagrangian submanifolds.
Mathematics Subject Classification: 53D12, 58D1.
Citation: Yuri Chekanov, Felix Schlenk. Notes on monotone Lagrangian twist tori. Electronic Research Announcements, 2010, 17: 104-121. doi: 10.3934/era.2010.17.104
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show all references

References:
[1]

Comm. Pure Appl. Math., 61 (2008), 1046-1051. doi: doi:10.1002/cpa.20216.  Google Scholar

[2]

Funkcional. Anal. i Prilozen., 1 (1967), 1-14. doi: doi:10.1007/BF01075861.  Google Scholar

[3]

J. Gökova Geom. Topol., 1 (2007), 51-91.  Google Scholar

[4]

in "New Perspectives and Challenges in Symplectic Field Theory," CRM Proc. Lecture Notes 49, AMS, (2009), 1-44.  Google Scholar

[5]

Geom. Topol., 13 (2009), 2881-2989. doi: doi:10.2140/gt.2009.13.2881.  Google Scholar

[6]

Math. Z., 223 (1996), 547-559.  Google Scholar

[7]

Duke Math. J., 95 (1998), 213-226. doi: doi:10.1215/S0012-7094-98-09506-0.  Google Scholar

[8]

Yu. Chekanov and F. Schlenk, Twist tori I: Construction and classification,, in preparation., ().   Google Scholar

[9]

Yu. Chekanov and F. Schlenk, Twist tori II: Non-displaceability,, in preparation., ().   Google Scholar

[10]

Int. Math. Res. Not., 35 (2004), 1803-1843. doi: doi:10.1155/S1073792804132716.  Google Scholar

[11]

Graduate Texts in Mathematics 150, Springer-Verlag, New York, 1995.  Google Scholar

[12]

in "Geometric Topology (Athens, GA, 1993)," AMS/IP Stud. Adv. Math. 2.1, AMS, (1997), 313-327.  Google Scholar

[13]

Ya. Eliashberg and L. Polterovich, Symplectic quasi-states on the quadric surface and Lagrangian submanifolds,, , ().   Google Scholar

[14]

Compos. Math., 145 (2009), 773-826. doi: doi:10.1112/S0010437X0900400X.  Google Scholar

[15]

J. Differential Geom., 28 (1988), 513-547.  Google Scholar

[16]

AMS/IP Studies in Advanced Mathematics 46.1, AMS, International Press, Somerville, MA, 2009.  Google Scholar

[17]

AMS/IP Studies in Advanced Mathematics 46.2, AMS, International Press, Somerville, MA, 2009.  Google Scholar

[18]

K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, Toric degeneration and non-displaceable Lagrangian tori in $S^2 \times S^2$,, \arXiv{1002.1660}., ().   Google Scholar

[19]

A. Gadbled, Exotic Hamiltonian tori in $\CP^2$ and $S^2 \times S^2$,, in preparation., ().   Google Scholar

[20]

Invent. math., 82 (1985), 307-347.  Google Scholar

[21]

Proc. Roy. Soc. Edinburgh Sect. A , 115 (1990), 25-38.  Google Scholar

[22]

Cambridge Studies in Advanced Mathematics 38, Cambridge University Press, Cambridge, 1994.  Google Scholar

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